Applying the first Angular Averaging Lemma REF  with \(j=sk\) , and invoking the elementary estimate
\(\big (\langle v \rangle ^{2}+\langle v_{*}\rangle ^{2}\big )^{sk} \le \big (\langle v\rangle ^{2s} + \langle v_{*}\rangle ^{2s}\big )^{k}\,,\quad s\in (0,1]\,,\) 
and using the exact combinatorial binomial expansion (REF ), applied with \(x=\langle v \rangle ^{2s}\)  and \(y=\langle v_{*}\rangle ^{2s}\) , yields estimate (REF ) for the weight function \(G_{sk}(v_*,v)\)  associated to the weak form of the Boltzmann flow. A detailed proof of this statement can also be found in [1]}, Lemma 2.
